Abstract

We describe a new procedure for retrieving both amplitude and phase of an optical beam from radial shearing measurements. Information from the sheared interferogram is used to estimate and improve the beam and wavefront shape in successive iterations. We present computer simulations and experimental results that show the performance of the method.

© 2008 Optical Society of America

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References

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  1. P. Hariharan and D. Sen, "Radial shearing interferometer," J. Sci. Instrum. 38, 428-432 (1961).
    [CrossRef]
  2. D. S. Brown, "Radial shear interferometry," J. Sci. Instrum. 39, 71-72 (1962).
    [CrossRef]
  3. M. V. R. K. Murty, "A compact radial shearing interferometer based on the law of refraction," Appl. Opt. 3, 853-857 (1964).
    [CrossRef]
  4. D. Malacara, Optical Shop Testing (Wiley, 1978).
  5. P. Hariharan, B. F. Oreb, and Z. Wan-Zhi, "Digital radial shearing interferometry: testing mirrors with a central hole," Opt. Acta 33, 251-254 (1986).
    [CrossRef]
  6. T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, "Radial shearing interferometer for inprocess measurement of diamond tuning," Opt. Eng. 39, 2696-2699 (2000).
    [CrossRef]
  7. T. Shirai, T. H. Barnes, and T. G. Haskell, "Adaptive wave-front correction by means of all-optical feedback interferometry," Opt. Lett. 25, 773-775 (2000).
    [CrossRef]
  8. D. C. L. Cheung, T. H. Barnes, and T. G. Haskell, "Feedback interferometry with membrane mirror for adaptive optics," Opt. Commun. 218, 33-41 (2003).
    [CrossRef]
  9. C. Y. Chung, K. C. Cho, C. C. Chang, C. H. Lin, W. C. Yen, and S. J. Chen, "Adaptive-optics system with liquid-crystal phase-shift interferometer," Appl. Opt. 45, 3409-3414 (2006).
    [CrossRef] [PubMed]
  10. W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, "Corneal topography measurement by means of radial shearing interference: part I--theoretical consideration," Optik 113, 39-45 (2002).
    [CrossRef]
  11. A. R. Barnes and I. C. Smith, "A combined phase, near field and far field diagnostic for large aperture laser systems," Proc. SPIE 3492, 564-572 (1999).
    [CrossRef]
  12. P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, "Wavefront and divergence of the beamlet prototype laser," Proc. SPIE 3492, 1019-1030 (1999).
    [CrossRef]
  13. T. M. Jeong, D.-K. Ko, and J. Lee, "Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers," Opt. Lett. 32, 232-234 (2007).
    [CrossRef] [PubMed]
  14. D. R. Kohler and V. L. Gamiz, "Interferogram reduction for radial-shear and local reference-holographic interferograms," Appl. Opt. 25, 1650-1652 (1986).
    [CrossRef] [PubMed]
  15. D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
    [CrossRef]
  16. D. Li, P. Wang, X. Li, H. Yang, and H. Chen, "Algorithm for near-field reconstruction based on radial-shearing interferometry," Opt. Lett. 30, 492-494 (2004).
    [CrossRef]
  17. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 72, 156-160 (1982).
    [CrossRef]
  18. K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, Vol. XXVI, E. Wolf, ed. (Elsevier, 1988), pp. 349-393.
    [CrossRef]

2007

2006

2004

2003

D. C. L. Cheung, T. H. Barnes, and T. G. Haskell, "Feedback interferometry with membrane mirror for adaptive optics," Opt. Commun. 218, 33-41 (2003).
[CrossRef]

2002

D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
[CrossRef]

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, "Corneal topography measurement by means of radial shearing interference: part I--theoretical consideration," Optik 113, 39-45 (2002).
[CrossRef]

2000

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, "Radial shearing interferometer for inprocess measurement of diamond tuning," Opt. Eng. 39, 2696-2699 (2000).
[CrossRef]

T. Shirai, T. H. Barnes, and T. G. Haskell, "Adaptive wave-front correction by means of all-optical feedback interferometry," Opt. Lett. 25, 773-775 (2000).
[CrossRef]

1999

A. R. Barnes and I. C. Smith, "A combined phase, near field and far field diagnostic for large aperture laser systems," Proc. SPIE 3492, 564-572 (1999).
[CrossRef]

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, "Wavefront and divergence of the beamlet prototype laser," Proc. SPIE 3492, 1019-1030 (1999).
[CrossRef]

1986

P. Hariharan, B. F. Oreb, and Z. Wan-Zhi, "Digital radial shearing interferometry: testing mirrors with a central hole," Opt. Acta 33, 251-254 (1986).
[CrossRef]

D. R. Kohler and V. L. Gamiz, "Interferogram reduction for radial-shear and local reference-holographic interferograms," Appl. Opt. 25, 1650-1652 (1986).
[CrossRef] [PubMed]

1982

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 72, 156-160 (1982).
[CrossRef]

1964

1962

D. S. Brown, "Radial shear interferometry," J. Sci. Instrum. 39, 71-72 (1962).
[CrossRef]

1961

P. Hariharan and D. Sen, "Radial shearing interferometer," J. Sci. Instrum. 38, 428-432 (1961).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 72, 156-160 (1982).
[CrossRef]

J. Sci. Instrum.

P. Hariharan and D. Sen, "Radial shearing interferometer," J. Sci. Instrum. 38, 428-432 (1961).
[CrossRef]

D. S. Brown, "Radial shear interferometry," J. Sci. Instrum. 39, 71-72 (1962).
[CrossRef]

Opt. Acta

P. Hariharan, B. F. Oreb, and Z. Wan-Zhi, "Digital radial shearing interferometry: testing mirrors with a central hole," Opt. Acta 33, 251-254 (1986).
[CrossRef]

Opt. Commun.

D. C. L. Cheung, T. H. Barnes, and T. G. Haskell, "Feedback interferometry with membrane mirror for adaptive optics," Opt. Commun. 218, 33-41 (2003).
[CrossRef]

Opt. Eng.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, "Radial shearing interferometer for inprocess measurement of diamond tuning," Opt. Eng. 39, 2696-2699 (2000).
[CrossRef]

D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
[CrossRef]

Opt. Lett.

Optik

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, "Corneal topography measurement by means of radial shearing interference: part I--theoretical consideration," Optik 113, 39-45 (2002).
[CrossRef]

Proc. SPIE

A. R. Barnes and I. C. Smith, "A combined phase, near field and far field diagnostic for large aperture laser systems," Proc. SPIE 3492, 564-572 (1999).
[CrossRef]

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, "Wavefront and divergence of the beamlet prototype laser," Proc. SPIE 3492, 1019-1030 (1999).
[CrossRef]

Other

D. Malacara, Optical Shop Testing (Wiley, 1978).

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, Vol. XXVI, E. Wolf, ed. (Elsevier, 1988), pp. 349-393.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Results of the numerical example: (a) signal phase under test, (b) retrieved wavefront, (c) signal irradiance under test, (d) retrieved irradiance. Note that on the plotted wavefronts we consider only the region where the signal irradiance exceeds 0.01.

Fig. 2
Fig. 2

Phase and irradiance rms error versus number of iterations corresponding to the example of Fig. 1 but for different magnification ratios.

Fig. 3
Fig. 3

Radial shearing experimental setup. P o , P i , object and image plane, respectively; M 1 , M 2 mirrors; BS 1 , BS 2 beam splitters; L 1 L 6 lenses.

Fig. 4
Fig. 4

Experimental results: (a) retrieved wavefront (considering a region where the signal irradiance exceeds 0.04), (b) retrieved wavefront along the camera's central horizontal and vertical lines, (c) retrieved beam irradiance, (d) measured beam irradiance. Note that the plot of Fig. 3(a) is rotated with respect to the plots of Figs. 3(c) and 3(d).

Fig. 5
Fig. 5

Difference between the measured and the retrieved CAM: (a) phase, (b) modulus.

Fig. 6
Fig. 6

(a) Measured and (b) retrieved background interference term.

Tables (1)

Tables Icon

Table 1 RMS Errors Corresponding to the Different Comparisons Between Experimental and Retrieved Data

Equations (8)

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I ( x , y ) = I s ( x , y ) + I r ( x , y ) + 2 I s I r cos ( Δ ϕ ( x , y ) + δ ) ,
2 I s I r cos ( Δ ϕ ( x , y ) + δ ) = u s ( x , y ) u s * ( x / M , y / M ) e i δ + u s * ( x , y ) u s ( x / M , y / M ) e i δ .
u 1 ( x , y ) = I CAM ( x , y ) ,
u k ( x , y ) = I CAM ( x , y ) / u k 1 * ( x / M , y / M ) .
u k ( x , y ) = I CAM ( x , y ) I CAM * ( x / M , y / M ) u k 2 ( x / M 2 , y / M 2 ) = u s ( x , y ) u s ( x / M 2 , y / M 2 ) u k 2 ( x / M 2 , y / M 2 ) = u s ( x , y ) u s * ( x / M 3 , y / M 3 ) u k 3 * ( x / M 3 , y / M 3 ) = = { u s ( x , y ) / u s ( x / M k , y / M k ) , k   even u s ( x , y ) u s * ( x / M k , y / M k ) , k   odd .
u k ( x , y ) = I CAM ( x , y ) / u k 1 * ( x / M + L , y / M ) .
u k ( x , y ) = u k 1 ( x / M , y / M ) I f ( x , y ) | u k 1 ( x / M , y / M ) | 2 + α ,
min { i , j = 1 N [ I b ( x i , y j ) c s I s ( x i , y j ) c r I r ( x i , y j ) ] 2 } ,

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